9 37

9_36

9_38

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 37's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 37 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8
Gauss code	-1, 4, -3, 1, -5, 6, -2, 9, -8, 3, -4, 2, -6, 5, -7, 8, -9, 7
Dowker-Thistlethwaite code	4 10 14 12 16 2 6 18 8
Conway Notation	[3,21,21]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 12, width is 5, Braid index is 5		[{11, 4}, {5, 3}, {4, 8}, {2, 5}, {7, 9}, {8, 6}, {3, 7}, {6, 1}, {10, 2}, {9, 11}, {1, 10}]

[edit Notes on presentations of 9 37]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 37"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8

In[5]:=

GaussCode[K]

Out[5]=

-1, 4, -3, 1, -5, 6, -2, 9, -8, 3, -4, 2, -6, 5, -7, 8, -9, 7

In[6]:=

DTCode[K]

Out[6]=

4 10 14 12 16 2 6 18 8

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[3,21,21]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(5,\{-1,-1,2,-1,-3,2,1,4,-3,2,-3,4\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 5, 12, 5 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{11, 4}, {5, 3}, {4, 8}, {2, 5}, {7, 9}, {8, 6}, {3, 7}, {6, 1}, {10, 2}, {9, 11}, {1, 10}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index $\{4,7\}$ Nakanishi index 2 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 10.9894 A-Polynomial See Data:9 37/A-polynomial

[edit Notes for 9 37's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant 0

[edit Notes for 9 37's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-11t+19-11t^{-1}+2t^{-2}$
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^4-3 z^2+1}
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 45, 0 }
Jones polynomial	$q^{4}-2q^{3}+5q^{2}-7q+7-8q^{-1}+7q^{-2}-4q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{4}+z^{4}a^{2}+z^{2}a^{2}+2a^{2}+z^{4}-z^{2}-2-2z^{2}a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+6az^{7}+3z^{7}a^{-1}+3a^{4}z^{6}+5a^{2}z^{6}+3z^{6}a^{-2}+5z^{6}+a^{5}z^{5}-6a^{3}z^{5}-13az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-8a^{4}z^{4}-17a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-13z^{4}-2a^{5}z^{3}+3a^{3}z^{3}+13az^{3}+6z^{3}a^{-1}-2z^{3}a^{-3}+5a^{4}z^{2}+14a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+12z^{2}-2a^{3}z-7az-5za^{-1}-2a^{2}+a^{-4}-2$
The A2 invariant	$-q^{16}+q^{14}+q^{12}-q^{10}+3q^{8}+q^{6}-3-2q^{-4}+q^{-6}+2q^{-8}-q^{-10}+q^{-12}+q^{-14}$
The G2 invariant	$q^{80}-2q^{78}+4q^{76}-7q^{74}+5q^{72}-4q^{70}-4q^{68}+16q^{66}-23q^{64}+29q^{62}-22q^{60}+6q^{58}+16q^{56}-38q^{54}+52q^{52}-49q^{50}+24q^{48}+8q^{46}-33q^{44}+50q^{42}-44q^{40}+24q^{38}+7q^{36}-27q^{34}+33q^{32}-28q^{30}-6q^{28}+40q^{26}-46q^{24}+41q^{22}-18q^{20}-17q^{18}+57q^{16}-71q^{14}+65q^{12}-48q^{10}+4q^{8}+43q^{6}-65q^{4}+65q^{2}-47+16q^{-2}+18q^{-4}-39q^{-6}+32q^{-8}-22q^{-10}-7q^{-12}+33q^{-14}-38q^{-16}+22q^{-18}+6q^{-20}-33q^{-22}+50q^{-24}-48q^{-26}+29q^{-28}-8q^{-30}-20q^{-32}+38q^{-34}-40q^{-36}+34q^{-38}-14q^{-40}+2q^{-42}+9q^{-44}-15q^{-46}+15q^{-48}-12q^{-50}+8q^{-52}-2q^{-54}-q^{-56}+3q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_37.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-q^{7}+3q^{5}-q^{3}-q-2q^{-3}+3q^{-5}-q^{-7}+q^{-9}$
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{32}-2 q^{30}-2 q^{28}+6 q^{26}-2 q^{24}-7 q^{22}+10 q^{20}+3 q^{18}-12 q^{16}+8 q^{14}+6 q^{12}-13 q^{10}+6 q^6-3 q^4-4 q^2+5+9 q^{-2} -8 q^{-4} -2 q^{-6} +13 q^{-8} -9 q^{-10} -7 q^{-12} +11 q^{-14} -3 q^{-16} -5 q^{-18} +4 q^{-20} - q^{-24} + q^{-26} }
3	$-q^{63}+2q^{61}+2q^{59}-3q^{57}-6q^{55}+2q^{53}+13q^{51}-q^{49}-19q^{47}-7q^{45}+26q^{43}+19q^{41}-25q^{39}-34q^{37}+22q^{35}+46q^{33}-8q^{31}-56q^{29}-5q^{27}+54q^{25}+15q^{23}-51q^{21}-30q^{19}+45q^{17}+32q^{15}-26q^{13}-33q^{11}+18q^{9}+34q^{7}+3q^{5}-32q^{3}-18q+26q^{-1}+32q^{-3}-20q^{-5}-51q^{-7}+10q^{-9}+56q^{-11}+4q^{-13}-58q^{-15}-11q^{-17}+49q^{-19}+24q^{-21}-38q^{-23}-25q^{-25}+25q^{-27}+21q^{-29}-10q^{-31}-16q^{-33}+4q^{-35}+8q^{-37}-2q^{-39}-4q^{-41}+q^{-43}+q^{-45}-q^{-49}+q^{-51}$
4	$q^{104}-2q^{102}-2q^{100}+3q^{98}+3q^{96}+6q^{94}-9q^{92}-13q^{90}+2q^{88}+10q^{86}+31q^{84}-8q^{82}-41q^{80}-26q^{78}+5q^{76}+82q^{74}+38q^{72}-48q^{70}-91q^{68}-69q^{66}+104q^{64}+137q^{62}+39q^{60}-122q^{58}-208q^{56}+11q^{54}+186q^{52}+197q^{50}-30q^{48}-287q^{46}-149q^{44}+110q^{42}+288q^{40}+120q^{38}-235q^{36}-241q^{34}-11q^{32}+262q^{30}+200q^{28}-122q^{26}-225q^{24}-87q^{22}+164q^{20}+187q^{18}-17q^{16}-167q^{14}-127q^{12}+61q^{10}+153q^{8}+85q^{6}-91q^{4}-160q^{2}-60+115q^{-2}+209q^{-4}+16q^{-6}-183q^{-8}-204q^{-10}+30q^{-12}+292q^{-14}+149q^{-16}-128q^{-18}-296q^{-20}-105q^{-22}+253q^{-24}+229q^{-26}+7q^{-28}-247q^{-30}-196q^{-32}+108q^{-34}+185q^{-36}+105q^{-38}-106q^{-40}-159q^{-42}-5q^{-44}+68q^{-46}+92q^{-48}-3q^{-50}-63q^{-52}-22q^{-54}+34q^{-58}+9q^{-60}-13q^{-62}-2q^{-64}-6q^{-66}+6q^{-68}+q^{-70}-3q^{-72}+2q^{-74}-2q^{-76}+q^{-78}-q^{-82}+q^{-84}$
5	$-q^{155}+2q^{153}+2q^{151}-3q^{149}-3q^{147}-3q^{145}+q^{143}+9q^{141}+13q^{139}-2q^{137}-20q^{135}-22q^{133}-7q^{131}+24q^{129}+49q^{127}+36q^{125}-30q^{123}-87q^{121}-77q^{119}+q^{117}+113q^{115}+163q^{113}+70q^{111}-122q^{109}-251q^{107}-195q^{105}+42q^{103}+322q^{101}+387q^{99}+126q^{97}-304q^{95}-569q^{93}-409q^{91}+149q^{89}+686q^{87}+733q^{85}+168q^{83}-652q^{81}-1042q^{79}-583q^{77}+432q^{75}+1201q^{73}+1037q^{71}-51q^{69}-1202q^{67}-1392q^{65}-396q^{63}+997q^{61}+1595q^{59}+828q^{57}-676q^{55}-1618q^{53}-1126q^{51}+330q^{49}+1462q^{47}+1286q^{45}+2q^{43}-1231q^{41}-1298q^{39}-210q^{37}+938q^{35}+1177q^{33}+372q^{31}-702q^{29}-1041q^{27}-424q^{25}+476q^{23}+875q^{21}+485q^{19}-293q^{17}-756q^{15}-533q^{13}+112q^{11}+662q^{9}+652q^{7}+95q^{5}-573q^{3}-803q-357q^{-1}+462q^{-3}+977q^{-5}+682q^{-7}-284q^{-9}-1125q^{-11}-1033q^{-13}+10q^{-15}+1166q^{-17}+1377q^{-19}+345q^{-21}-1089q^{-23}-1605q^{-25}-733q^{-27}+812q^{-29}+1696q^{-31}+1087q^{-33}-457q^{-35}-1549q^{-37}-1305q^{-39}+15q^{-41}+1246q^{-43}+1358q^{-45}+342q^{-47}-825q^{-49}-1191q^{-51}-576q^{-53}+399q^{-55}+907q^{-57}+640q^{-59}-75q^{-61}-579q^{-63}-543q^{-65}-127q^{-67}+285q^{-69}+390q^{-71}+185q^{-73}-99q^{-75}-223q^{-77}-155q^{-79}-2q^{-81}+105q^{-83}+96q^{-85}+29q^{-87}-36q^{-89}-49q^{-91}-20q^{-93}+9q^{-95}+16q^{-97}+11q^{-99}+4q^{-101}-8q^{-103}-5q^{-105}+2q^{-107}-q^{-109}+3q^{-113}-q^{-115}-2q^{-117}+q^{-119}-q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+q^{12}-q^{10}+3q^{8}+q^{6}-3-2q^{-4}+q^{-6}+2q^{-8}-q^{-10}+q^{-12}+q^{-14}$
1,1	$q^{44}-4q^{42}+10q^{40}-22q^{38}+42q^{36}-70q^{34}+100q^{32}-136q^{30}+169q^{28}-186q^{26}+186q^{24}-156q^{22}+117q^{20}-40q^{18}-44q^{16}+138q^{14}-229q^{12}+284q^{10}-348q^{8}+352q^{6}-343q^{4}+300q^{2}-214+146q^{-2}-42q^{-4}-30q^{-6}+104q^{-8}-154q^{-10}+166q^{-12}-170q^{-14}+152q^{-16}-132q^{-18}+99q^{-20}-72q^{-22}+50q^{-24}-32q^{-26}+21q^{-28}-10q^{-30}+6q^{-32}-2q^{-34}+q^{-36}$
2,0	$q^{42}-q^{40}-2q^{38}+3q^{34}+q^{32}-6q^{30}+8q^{26}+5q^{24}-6q^{22}-2q^{20}+8q^{18}+4q^{16}-9q^{14}-5q^{12}+2q^{10}-5q^{8}-4q^{6}+2q^{2}+2+8q^{-2}+6q^{-4}-3q^{-6}-q^{-8}+8q^{-10}-11q^{-14}+7q^{-18}-8q^{-22}-3q^{-24}+4q^{-26}+2q^{-28}-q^{-30}+q^{-34}+q^{-36}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+2q^{28}-6q^{26}+4q^{24}+6q^{22}-6q^{20}+7q^{18}+8q^{16}-11q^{14}+2q^{10}-9q^{8}-2q^{6}+4q^{4}+6q^{2}+1+q^{-2}+8q^{-4}-5q^{-6}-10q^{-8}+8q^{-10}-4q^{-12}-8q^{-14}+9q^{-16}-3q^{-20}+4q^{-22}+q^{-24}-q^{-26}+q^{-28}$
1,0,0	$-q^{21}+q^{19}+q^{15}-q^{13}+3q^{11}+q^{9}+2q^{7}-2q-3q^{-1}-2q^{-5}+q^{-7}+2q^{-11}-q^{-13}+q^{-15}+q^{-17}+q^{-19}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-4q^{30}+6q^{28}-8q^{26}+10q^{24}-10q^{22}+10q^{20}-7q^{18}+6q^{16}+q^{14}-4q^{12}+12q^{10}-15q^{8}+18q^{6}-20q^{4}+18q^{2}-19+11q^{-2}-8q^{-4}+q^{-6}+2q^{-8}-6q^{-10}+10q^{-12}-10q^{-14}+11q^{-16}-8q^{-18}+7q^{-20}-4q^{-22}+3q^{-24}-q^{-26}+q^{-28}$
1,0	$q^{56}-2q^{52}-2q^{50}+2q^{48}+4q^{46}-2q^{44}-7q^{42}-2q^{40}+9q^{38}+8q^{36}-4q^{34}-9q^{32}+2q^{30}+12q^{28}+7q^{26}-9q^{24}-9q^{22}+2q^{20}+7q^{18}-4q^{16}-10q^{14}-2q^{12}+6q^{10}+2q^{8}-6q^{6}-q^{4}+8q^{2}+9-3q^{-2}-5q^{-4}+4q^{-6}+9q^{-8}-2q^{-10}-11q^{-12}-5q^{-14}+8q^{-16}+7q^{-18}-7q^{-20}-11q^{-22}+10q^{-26}+4q^{-28}-4q^{-30}-5q^{-32}+q^{-34}+4q^{-36}+2q^{-38}-q^{-40}-q^{-42}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+4q^{76}-7q^{74}+5q^{72}-4q^{70}-4q^{68}+16q^{66}-23q^{64}+29q^{62}-22q^{60}+6q^{58}+16q^{56}-38q^{54}+52q^{52}-49q^{50}+24q^{48}+8q^{46}-33q^{44}+50q^{42}-44q^{40}+24q^{38}+7q^{36}-27q^{34}+33q^{32}-28q^{30}-6q^{28}+40q^{26}-46q^{24}+41q^{22}-18q^{20}-17q^{18}+57q^{16}-71q^{14}+65q^{12}-48q^{10}+4q^{8}+43q^{6}-65q^{4}+65q^{2}-47+16q^{-2}+18q^{-4}-39q^{-6}+32q^{-8}-22q^{-10}-7q^{-12}+33q^{-14}-38q^{-16}+22q^{-18}+6q^{-20}-33q^{-22}+50q^{-24}-48q^{-26}+29q^{-28}-8q^{-30}-20q^{-32}+38q^{-34}-40q^{-36}+34q^{-38}-14q^{-40}+2q^{-42}+9q^{-44}-15q^{-46}+15q^{-48}-12q^{-50}+8q^{-52}-2q^{-54}-q^{-56}+3q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 37"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$2t^{2}-11t+19-11t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^4-3 z^2+1}

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 45, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{4}-2q^{3}+5q^{2}-7q+7-8q^{-1}+7q^{-2}-4q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{2}a^{4}+z^{4}a^{2}+z^{2}a^{2}+2a^{2}+z^{4}-z^{2}-2-2z^{2}a^{-2}+a^{-4}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+6az^{7}+3z^{7}a^{-1}+3a^{4}z^{6}+5a^{2}z^{6}+3z^{6}a^{-2}+5z^{6}+a^{5}z^{5}-6a^{3}z^{5}-13az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-8a^{4}z^{4}-17a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-13z^{4}-2a^{5}z^{3}+3a^{3}z^{3}+13az^{3}+6z^{3}a^{-1}-2z^{3}a^{-3}+5a^{4}z^{2}+14a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+12z^{2}-2a^{3}z-7az-5za^{-1}-2a^{2}+a^{-4}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n100,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 37"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $2t^{2}-11t+19-11t^{-1}+2t^{-2}$, $q^{4}-2q^{3}+5q^{2}-7q+7-8q^{-1}+7q^{-2}-4q^{-3}+3q^{-4}-q^{-5}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11n100,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-3, -1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-12$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -8} $72$ $82$ $22$ $96$ ${\frac {496}{3}}$ ${\frac {160}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 24} $-288$ $32$ $-984$ $-264$ $-{\frac {8191}{10}}$ $-{\frac {974}{15}}$ $-{\frac {1954}{5}}$ ${\frac {85}{2}}$ $-{\frac {671}{10}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 9 37. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 1   -1 5               4 1   3 3             3 1     -2 1           4 4       0 -1         5 4         -1 -3       2 3           -1 -5     2 5             3 -7   1 2               -1 -9   2                 2 -11 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{12}-2q^{11}+q^{10}+5q^{9}-11q^{8}+3q^{7}+19q^{6}-29q^{5}+q^{4}+41q^{3}-44q^{2}-5q+58-48q^{-1}-14q^{-2}+59q^{-3}-39q^{-4}-20q^{-5}+46q^{-6}-20q^{-7}-18q^{-8}+26q^{-9}-5q^{-10}-11q^{-11}+9q^{-12}-3q^{-14}+q^{-15}$
3	$q^{24}-2q^{23}+q^{22}+q^{21}+q^{20}-7q^{19}+3q^{18}+11q^{17}-3q^{16}-27q^{15}+9q^{14}+42q^{13}+q^{12}-77q^{11}-4q^{10}+104q^{9}+26q^{8}-137q^{7}-51q^{6}+166q^{5}+78q^{4}-183q^{3}-112q^{2}+197q+130-189q^{-1}-156q^{-2}+183q^{-3}+165q^{-4}-158q^{-5}-172q^{-6}+132q^{-7}+172q^{-8}-100q^{-9}-159q^{-10}+57q^{-11}+151q^{-12}-34q^{-13}-120q^{-14}-2q^{-15}+100q^{-16}+14q^{-17}-66q^{-18}-26q^{-19}+44q^{-20}+23q^{-21}-22q^{-22}-19q^{-23}+11q^{-24}+11q^{-25}-4q^{-26}-5q^{-27}+3q^{-29}-q^{-30}$
4	$q^{40}-2q^{39}+q^{38}+q^{37}-3q^{36}+5q^{35}-7q^{34}+5q^{33}+6q^{32}-15q^{31}+9q^{30}-18q^{29}+27q^{28}+31q^{27}-49q^{26}-13q^{25}-59q^{24}+87q^{23}+126q^{22}-73q^{21}-86q^{20}-213q^{19}+140q^{18}+337q^{17}+7q^{16}-163q^{15}-517q^{14}+89q^{13}+591q^{12}+229q^{11}-139q^{10}-875q^{9}-102q^{8}+759q^{7}+506q^{6}+4q^{5}-1137q^{4}-336q^{3}+780q^{2}+705q+197-1231q^{-1}-511q^{-2}+680q^{-3}+774q^{-4}+373q^{-5}-1163q^{-6}-603q^{-7}+492q^{-8}+734q^{-9}+523q^{-10}-959q^{-11}-626q^{-12}+241q^{-13}+596q^{-14}+626q^{-15}-637q^{-16}-564q^{-17}-32q^{-18}+366q^{-19}+632q^{-20}-282q^{-21}-396q^{-22}-210q^{-23}+107q^{-24}+494q^{-25}-25q^{-26}-169q^{-27}-221q^{-28}-68q^{-29}+275q^{-30}+61q^{-31}-8q^{-32}-123q^{-33}-101q^{-34}+102q^{-35}+39q^{-36}+35q^{-37}-37q^{-38}-57q^{-39}+25q^{-40}+8q^{-41}+20q^{-42}-4q^{-43}-18q^{-44}+4q^{-45}+5q^{-47}-3q^{-49}+q^{-50}$
5	$q^{60}-2q^{59}+q^{58}+q^{57}-3q^{56}+q^{55}+5q^{54}-5q^{53}+4q^{51}-10q^{50}-2q^{49}+17q^{48}+2q^{47}+5q^{46}-3q^{45}-39q^{44}-31q^{43}+30q^{42}+67q^{41}+72q^{40}+6q^{39}-146q^{38}-184q^{37}-38q^{36}+191q^{35}+356q^{34}+211q^{33}-251q^{32}-596q^{31}-454q^{30}+155q^{29}+860q^{28}+926q^{27}+16q^{26}-1104q^{25}-1429q^{24}-460q^{23}+1226q^{22}+2093q^{21}+1032q^{20}-1216q^{19}-2660q^{18}-1780q^{17}+982q^{16}+3185q^{15}+2576q^{14}-607q^{13}-3544q^{12}-3325q^{11}+110q^{10}+3701q^{9}+4010q^{8}+425q^{7}-3755q^{6}-4481q^{5}-933q^{4}+3609q^{3}+4851q^{2}+1391q-3460-4996q^{-1}-1752q^{-2}+3163q^{-3}+5081q^{-4}+2059q^{-5}-2903q^{-6}-4986q^{-7}-2302q^{-8}+2518q^{-9}+4858q^{-10}+2522q^{-11}-2125q^{-12}-4596q^{-13}-2701q^{-14}+1618q^{-15}+4241q^{-16}+2861q^{-17}-1051q^{-18}-3791q^{-19}-2940q^{-20}+470q^{-21}+3153q^{-22}+2928q^{-23}+182q^{-24}-2507q^{-25}-2764q^{-26}-662q^{-27}+1697q^{-28}+2436q^{-29}+1124q^{-30}-1003q^{-31}-1997q^{-32}-1260q^{-33}+304q^{-34}+1440q^{-35}+1314q^{-36}+148q^{-37}-909q^{-38}-1096q^{-39}-465q^{-40}+425q^{-41}+855q^{-42}+538q^{-43}-89q^{-44}-531q^{-45}-512q^{-46}-112q^{-47}+297q^{-48}+378q^{-49}+176q^{-50}-101q^{-51}-251q^{-52}-177q^{-53}+17q^{-54}+141q^{-55}+120q^{-56}+28q^{-57}-59q^{-58}-84q^{-59}-33q^{-60}+29q^{-61}+42q^{-62}+18q^{-63}-2q^{-64}-18q^{-65}-20q^{-66}+4q^{-67}+11q^{-68}+3q^{-69}-5q^{-72}+3q^{-74}-q^{-75}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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